Calibrating method and system, recording medium for this method

ABSTRACT

This method for calibrating coefficients of an observer of a variable state comprises:
     the measurement ( 50 ) of a variable z k  of the physical system at N different instants, this variable z k  being a function of the state variable x k ,   the determining ( 54 ) of a vector p of coefficients which minimizes the following criterion in complying with a predetermined set Δ of constraints:   

     
       
         
           
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         {circumflex over (x)} k  is the estimation of the variable x k  built by the observer calibrated with the coefficients of the vector p at the instant k,   φ is a known function which links the estimation {circumflex over (x)} k  to an estimation {circumflex over (z)} k  of the variable z k ,   ∥ . . . ∥ 2  is a norm, and   the constraint or the constraints of the set Δ dictate that the trajectory of the variable z k  should be included in a corridor of uncertainty situated on either side of the trajectory of the estimation {circumflex over (z)} k  at least for the majority of the instants k.

The invention pertains to a method and system for calibratingcoefficients of an observer of a variable state x_(k) of a physicalsystem. The invention also pertains to a recording medium to implementthis method.

An observer estimates the state variable x_(k) from measurements y_(k)of physical quantities or magnitudes of this system where the index kidentifies the instant of measurement.

There are known ways of calibrating the coefficients of an observer bysuccessive experiments with different sets of coefficients until one setof coefficients gives the desired operation by this observer. Typically,the behavior desired for the observer is:

-   -   fast convergence towards the most exact possible estimation        {circumflex over (x)}_(k) of the variable x_(k), and    -   high stability.

This calibration by experiment is part of a heuristic approach. Thecalibration is often lengthy, and it is difficult to be certain that thebest possible calibration of the observer has been obtained.

To resolve this problem, other alternative methods have been proposed.For example, the patent application CA 2 226 282 describes a method forcalibrating coefficients of a Kalman filter by means of a neuralnetwork. However, these alternative methods are often as complex toimplement as the method that uses experimentation.

The invention is therefore aimed at proposing a method of calibrationthat is simpler to implement.

Thus, an object of the invention is a method for calibratingcoefficients of an observer comprising:

-   -   the measurement of a variable z_(k) of the physical system at N        different instants, this variable z_(k) being a function of the        state variable x_(k);    -   the determining of a vector p of coefficients which minimizes        the following criterion in complying with a predetermined set Δ        of constraints:

$\{ {{f(p)} = {\sum\limits_{k = 1}^{N}\; {{z_{k} - {\varphi ( {{\hat{x}}_{k},p} )}}}^{2}}} \}$

where:

-   -   {circumflex over (x)}_(k) is the estimation of the variable        x_(k) built by the observer calibrated with the coefficients of        the vector p at the instant k;    -   φ is a known function which links the estimation {circumflex        over (x)}_(k) to an estimation {circumflex over (z)}_(k) of the        variable z_(k);    -   ∥ . . . ∥² is a norm of the difference between z_(k) and        φ({circumflex over (x)}_(k),p), and    -   the constraint or the constraints of the set Δ dictate that the        trajectory of the variable z_(k) should be included in a        corridor of uncertainty situated on either side of the        trajectory of the estimation {circumflex over (z)}_(k) at least        for the majority of the instants k.

The above method is used to determine the coefficients enabling anefficient functioning of the observer in a simple way and withoutrecourse to different experiments. Furthermore, the coefficients thusdetermined make it possible to ensure that the criterion f(p) has beenminimized.

Finally, the use of the set Δ takes account of the fact that there is anuncertainty on the estimation {circumflex over (z)}_(k). Taking accountof this uncertainty when determining the vector p that minimizes thecriterion f(p) gives better calibration than if this uncertainty werenot taken into account. In particular, aberrant approaches which couldminimize the criterion f(p) are eliminated.

The embodiments of this calibration method may include one or more ofthe following characteristics:

-   -   the set Δ includes a constraint which dictates that the variable        z_(k) should be included between {circumflex over (z)}_(k)−a and        {circumflex over (z)}_(k)+b, where a and b are predetermined        positive constants;    -   the constants a and b are proportional to a mean square        deviation σ_({grave over (z)}) _(k) representing the uncertainty        in the estimation {circumflex over (z)}_(k);    -   the set Δ of constraints includes a constraint which dictates        that the mean of the deviations between the variable z_(k) and        its estimation {circumflex over (z)}_(k) should be below a        predetermined threshold S₁;    -   the threshold S₁ is a function of the uncertainty on the        estimation {circumflex over (z)}_(k);    -   the observer is a Kalman filter, an extended Kalman filter or an        Unscented Kalman Filter and the coefficients to be calibrated        are coefficients of the covariance matrices R and Q of this        filter;    -   the R and Q matrices are chosen to be diagonal matrices.

These embodiments of the calibration method furthermore have thefollowing advantages:

-   -   dictating that the variable z_(k) should be included between        {circumflex over (z)}_(k)−a and {circumflex over (z)}_(k)+b        reduces the space of search in which the vector p must be        sought, thus accelerating the execution of this method;    -   using constants a and b proportional to the mean square        deviation σ_({circumflex over (z)}) _(k) improves the        calibration,    -   using a constraint which dictates that the mean of the        deviations in terms of absolute value between the variable z_(k)        and its estimation {circumflex over (z)}_(k) should be below the        threshold S₁ relaxes the dictated constraints thus facilitating        the determination of a vector p;    -   using a threshold S₁ which is a function of the uncertainty of        the estimation of {circumflex over (z)}_(k) improves the        calibration;    -   choosing diagonal R and Q matrices decreases the number of        coefficients to be determined, thus facilitating the        calibration.

An object of the invention is also an information-recording mediumcomprising instructions to implement the above method when theinstructions are executed by a programmable electronic computer.

An object of the invention is also a system for calibrating coefficientsof an observer of a state variable x_(k) of a physical system frommeasurements y_(k) of physical quantities of this system where the indexk identifies the instant of measurement, the system comprising:

-   -   at least one sensor capable of measuring a variable z_(k) of the        physical system at N different instants, this variable z_(k)        being a function of the state variable x_(k), and    -   a computer capable of determining a vector p of coefficients        which minimizes the following criterion in complying with a        predetermined set Δ of constraints:

$\{ {{f(p)} = {\sum\limits_{k = 1}^{N}\; {{z_{k} - {\varphi ( {{\hat{x}}_{k},p} )}}}^{2}}} \}$

where:

-   -   {circumflex over (x)}_(k) is the estimation of the variable        x_(k) built by the observer calibrated with the coefficients of        the vector p at the instant k,    -   φ is a known function which links the estimation {circumflex        over (x)}_(k) to an estimation {circumflex over (z)}_(k) of the        variable z_(k),    -   ∥ . . . ∥² is a norm of the difference between z_(k) and        φ({circumflex over (x)}_(k),p), and    -   the constraint or the constraints of the set Δ dictate that the        trajectory of the variable z_(k) should be included in a        corridor of uncertainty situated on either side of the        trajectory of the estimation {circumflex over (z)}_(k) at least        for the majority of the instants k.

The invention shall be understood more clearly from the followingdescription, given purely by way of a non-exhaustive example and madewith reference to the appended drawings of which:

FIG. 1 is a schematic illustration of the architecture of a localizingsystem including an observer and a system for calibrating this observer,

FIG. 2 is a flowchart of a method of calibration of the observer of thesystem of FIG. 1, and

FIG. 3 is a graph illustrating the trajectory of a variable z_(k) andits estimation {circumflex over (z)}_(k).

Here below in this description, the characteristics and functions wellknown to those skilled in the art shall not be described in greaterdetail.

Here, a trajectory designates the variations in time of a variable. Thevariable that varies in time may be any unspecified variable. Inparticular, this variable is not necessarily a position in space. Forexample, it may a temperature or any other physical quantity whichvaries over time.

FIG. 1 represents a physical system, i.e. in this case an object 4 and asystem 8 for localizing this object 4 in the reference system 8.

For example, the object 4 is a probe or a catheter introduced into ahuman body. The object 4 is mobile in the referential system 8.

The referential system 8 is a fixed referential system having threeorthonormal axes X, Y and Z.

The localization of the object 4 in the referential system 8 consists infinding its x_(p), y_(p), z_(p) position and its θ_(x), θ_(y) and θ_(z)orientation. The angles θ_(x), θ_(y) and θ_(z) represent the orientationof the object 4 respectively in relation to the axes X, Y and Z.

To localize the object 4, the system 6 herein comprises magnetic fieldsources and magnetic field sensors some of which are linked to thereferential system 8 while others are fixed without any degree offreedom to the object 4 to be localized. Here, the magnetic field sensor10 is linked without any degree of freedom to the object 4. The sensor10 is for example a triaxial sensor, i.e. a sensor capable of measuringthe projection of the magnetic field on three non-colinear measurementaxes. Such a sensor measures the direction of the magnetic field. Moregenerally, this sensor also measures the amplitude of the magneticfield.

For example, the measurement axes are mutually orthogonal. These axesare fixedly linked to the object 4.

This sensor 10 is connected by means of a flexible wire link to aprocessing unit 12.

The unit 12 is also connected to three sources 14 to 16 of the magneticfield. These sources are for example magnetic field triaxial sources. Amagnetic field triaxial source emits a magnetic field, along threemutually non-colinear emission axes. For example, such a source isformed by several aligned magnetic field uniaxial emitters, respectivelyon each of the emission axes of the source. The uniaxial emitter mainlyemits the magnetic field along a single axis. For example, it is a coilwhose turns are wound about a same emission axis. In this case, theemission axis coincides with the winding axis of the turns.

Here, the sources 14 to 16 are identical to each other. These sourcesare fixed in the referential system 8.

The processing unit 12 powers the sources 14 to 16 with alternatingcurrent to generate the magnetic field measured by the sensor 10. Theunit 12 also acquires the measurements of the magnetic field made by thesensor 10.

The unit 12 is equipped with an observer 18 which localizes the object 4in the referential system 8 from the measurements acquired. Here, theobserver 18 determines the position and orientation of the object 4 byresolving a system of equations. This system of equations is obtained bymodeling the magnetic interactions between the sources 14 to 16 and thesensor 10. In this system of equations, the position x_(p), y_(p) andz_(p) and the orientations θ_(x), θ_(y) and θ_(z) of the object 4 arethe unknown quantities while the values of the other parameters areobtained from measurements made by the sensor 10. Further information onsuch systems of equations may for example be found in the patentapplication EP 1 502 544.

Here, the observer 18 is a Kalman filter.

This observer 18 builds an estimation {circumflex over (x)}_(k) of astate variable x_(k) from a measurement y_(k). The variable x_(k) is avector containing the position x_(p), y_(p), z_(p) of the object 4 andits orientation θ_(x), θ_(y) and θ_(z) in the orthonormal referentialsystem 8, the measurement y_(k) is a measurement vector comprising themeasurements made along the three measurement axes of the sensor 10 atan instant k.

The equations of the Kalman filter of the observer 18 are the following:

$\begin{matrix}\begin{Bmatrix}{{\hat{x}}_{k + {1/k}} = {F_{k}{\hat{x}}_{k}}} \\{P_{k + {1/k}} = {{F_{k}P_{k/k}F_{k}^{T}} + Q_{k}}}\end{Bmatrix} & (1) \\\begin{Bmatrix}{{\hat{x}}_{k + {1/k} + 1} = {{\hat{x}}_{k + {1/k}} + {K_{k + 1}( {y_{k + 1} - {C_{k + 1}{\hat{x}}_{k + {1/k}}}} )}}} \\{P_{k + {1/k} + 1} = {P_{k + {1/k}} - {K_{k + 1}C_{k + 1}P_{k + {1/k}}}}}\end{Bmatrix} & (2)\end{matrix}$

where:

-   -   {circumflex over (x)}_(k+1/k) is the estimation of the variable        x_(k) obtained solely from data available at the instant k,    -   F_(k) is the matrix linking the estimation {circumflex over        (x)}_(k) to the estimation {circumflex over (x)}_(k+1/k),    -   P_(k/k) is the matrix for estimating the covariance, of the        errors on the estimation {circumflex over (x)}_(k),    -   Q_(k) is a matrix of covariance of the noise of the equation of        the model containing inter alia the modeling errors,    -   {circumflex over (x)}_(k+1/k+1) is the estimation {circumflex        over (x)}_(k+1) corrected at the instant k+1 from the        measurements made at the instant k+1,    -   y_(k) is the vector of the measurements made at the instant k,    -   C_(k) is the matrix that links the variable x_(k) to the        measurement y_(k),    -   P_(k+1/k+1) is the matrix of covariance of the error of        estimation {circumflex over (x)}_(k) corrected on the basis of        the measurements y_(k),    -   K_(k+1) is the gain of the Kalman filter.

The matrix F_(k) is for example obtained by modelizing the magneticinteractions between the sources 14 to 16 and the sensor 10 on the basisof the laws of electromagnetism.

The gain K_(k+1) of the Kalman filter is given by the followingrelationship:

K _(k+1) =P _(k+1/k) C ^(T) _(k+1) [C _(k+1) P _(k+1/k) C ^(T) _(k+1) +R_(k+1)]⁻¹  (3)

where R_(k+1) is the matrix of covariance of the noise in themeasurement y_(k).

The given equations of the Kalman filter are classic and are thereforenot described in greater detail herein.

The use of a Kalman filter gives a precise estimation of the positionand orientation of the object 4. However, before this, the matricesR_(k) and Q_(k) have to be calibrated, i.e. the values of theircoefficients have to be determined. To simplify the description, it isassumed here that the values of these coefficients are constant. Thematrices R_(k) and Q_(k) are therefore also denoted as R and Q herebelow.

Here p_(R) and p_(Q) denote the set of coefficients respectively of thematrices R and Q to be determined.

Once the Kalman filter has been calibrated, these sets P_(R) and P_(Q)are registered in a memory 20 connected to the unit 12 so that they canbe used by the observer 18 to estimate the position of the object 4.

FIG. 1 also represents a system 30 for calibrating the observer 18, thissystem automatically determines the sets p_(R) and p_(Q) of coefficientsof the matrices R and Q of the observer 18.

To this end, the system 30 is equipped with one or more sensors 32 whichmeasure a variable z_(k). This variable z_(k) is a variable whose valueis a function of the variable x_(k). More specifically, the variablez_(k) is connected to the variable x_(k) by a known function φ. Thisfunction φ may be a linear function or on the contrary a non-linearfunction. For example, the sensor 32 measures the acceleration along theX, Y and Z axes of the object 4. In this case, the sensor 32 is formedby several accelerometers.

In another embodiment, the sensor 32 can directly measure the positionof the object 4 with sensors other than those used to carry out themeasurement y_(k). For example, the sensor 32 may, to this end, haveseveral cameras or other sensors and other magnetic field sources tomeasure the position of the object 4. In this particular case, thefunction φ is the identity function.

The system 30 also has a computer 34 capable of acquiring themeasurements of the sensor 32. This computer 34 is also capable ofcalibrating the observer automatically by determining the set ofcoefficients P_(R) and P_(Q) from the measurements made by the sensor32.

In the particular case shown in FIG. 1, the computer 34 is connected tothe memory 20 so that it can directly record the set of coefficientsp_(R) and p_(Q) in this memory 20.

The computer 34 is made with a programmable electronic computer capableof executing the instructions recorded in an information-recordingmedium. Here, the computer 34 is connected to a memory 36 and thismemory 36 contains the instructions needed to execute the method of FIG.2.

The working of the calibration system 30 shall now be described ingreater detail with reference to the method of FIG. 2 and the graph ofFIG. 3.

Initially, at a step 40, the relationship φ which connects the variablez_(k) to the variable x_(k) is set up. For example, to this end, thelaws of physics are used.

Then, at a step 42, the number of coefficients to be used to calibratethe observer 18 is determined. To this end, knowledge on the physicalworking of the system 6 is used. Furthermore, this step applies theprinciple of parsimony according to which the number of coefficients tobe calibrated is limited to the greatest possible extent.

For example, in the particular case described herein, it is assumed thatthe measurements made on each of the measurement axes are independent ofone another. Thus, the matrix R is taken to be a diagonal matrix.Similarly, it is assumed that the errors in the state equation areindependent. Thus, the matrix Q is taken to be a diagonal matrix. Theytherefore have the following form:

R=diag(p ¹ _(R) , . . . ,p ^(l) _(R)),p ^(i) _(R)≧0,i=1:l  (7)

Q=diag(p ¹ _(Q) , . . . ,p ^(n) _(Q)),p ^(i) _(Q)≧0,i=1:n  (8)

where:

-   -   the index i represents the index of the coefficient in the set        of coefficients p_(R) or p_(Q),    -   l and n are respectively the dimensions of the matrices R and Q.

Furthermore, the matrices R and Q are symmetrical matrices defined asbeing positive, i.e. all matrices of which all the Eigen values arepositive. To ensure this condition, the matrices R and Q can be writtenas noted here below:

R=X^(T)X and Q=Y^(T)Y, where the matrices X and Y are diagonal matrices.It can be noted that when the matrices R and Q are any unspecifiedmatrices, the matrices X and Y are higher triangular matrices.

At the end of the step 42, with the assumptions made, the vector p ofcoefficients to be determined is defined by the following relationship:

p=[p ^(T) _(R) ,p ^(T) _(Q)]^(T)  (9)

At a step 44, the criterion to be optimized to determine the vector p ischosen. Here, this vector is given by the following relationship:

$\begin{matrix}{\min\limits_{p \in \Delta}\{ {{f(p)} = {\sum\limits_{k = 1}^{N}\; {{z_{k} - {\varphi ( {{\hat{x}}_{k},p} )}}}_{2}^{2}}} \}} & (10)\end{matrix}$

where:

-   -   ∥ . . . ∥₂ represents the L₂ norm operation corresponding to the        Euclidian norm,    -   <<min>> represents the operation of finding the vector p which        minimizes the function f(p),    -   Δ is the set of constraints that must be verified by the vector        p which minimizes the function f(p).

The function φ which appears in the criterion (10) is the same as theone introduced here above. This function φ is used to obtain theestimation {circumflex over (z)}_(k) of the variable z_(k) from theestimation {circumflex over (x)}_(k) as indicated in the followingrelationship:

{circumflex over (z)} _(k)=φ({circumflex over (x)} _(k) ,p)  (11)

It will be noted that the function φ also depends on the vector p sincethe estimation {circumflex over (x)}_(k) which constitutes the argumentof the function φ is itself a function of the vector p. For this reason,in the above relationship, the estimation {circumflex over (x)}_(k) andthe vector p are indicated as the argument of the function φ.

At a step 46, the set Δ of constraints to be met by the vector p isdefined. This set Δ comprises at least one constraint which dictatesthat estimation {circumflex over (z)}_(k) and the variable z_(k) shouldbe proximate to each other. More specifically, this constraint hereindictates that the trajectory of the variable z_(k) should be included,at least on an average, in a corridor 48 of uncertainty (FIG. 3)situated on either side of the trajectory of estimation {circumflex over(z)}_(k).

This corridor 48 is demarcated by two boundaries L_(min) and L_(max)respectively situated on each side of the trajectory of estimation{circumflex over (z)}_(k). The corridor of uncertainty represents thefact that the estimation {circumflex over (z)}_(k) is only known with anuncertainty attached to it. This uncertainty comes from the fact thatthere is uncertainty in the estimation {circumflex over (x)}_(k) used tobuild the estimation {circumflex over (z)}_(k). Thus, to calibrate theobserver 18 properly, it is necessary that, for the majority of theinstants k, the variable z_(k) should be included in the corridor 48.

In the particular case shown in FIG. 3, the boundaries L_(max) andL_(min) which demarcate the corridor 48 are separated from thetrajectory of {circumflex over (z)}_(k), and respectively by distances aand b. In the case of FIG. 3, these distances a and b are constant inthe course of time.

Preferably, the distances a and b are a function of the uncertainty inthe estimation {circumflex over (z)}_(k). For example, the distances aand b are all the greater as the mean square deviationσ_({circumflex over (z)}) _(k) is great. σ_({circumflex over (z)}) _(k)is the mean square deviation of the error in the estimation {circumflexover (z)}_(k). For example, here, the set Δ includes a constraint whichdictates that the mean of the divergences between the variable z_(k) andthe estimation {circumflex over (z)}_(k) should be below a predeterminedthreshold S₁. This constraint can be expressed by means of the followingrelationship:

$\begin{matrix}{\sqrt{\sum\limits_{k = 1}^{N}{{z_{k} - {\hat{z}}_{k}}}_{2}} \leq S_{1}} & (11)\end{matrix}$

The threshold S1 is for example defined by the following relationship:

S ₁ =√{square root over (N)}ασ _({circumflex over (z)}) _(k)   (12)

where:

-   -   N is the number of instants k, and    -   α is a constant, for example equal to three.

For each vector p potentially minimizing the criterion (10), it ispossible to determine the mean square deviationσ_({circumflex over (z)}) _(k) . Indeed, when a vector p is known, thematrices R and Q are also known. From the matrices R and Q, it ispossible to compute the matrix P_(k/k) of the covariance of the error inthe estimation of {circumflex over (x)}_(k). From this matrix P_(k/k) itis possible to determine the uncertainty affecting the estimation of{circumflex over (x)}_(k). With this uncertainty being known, it is alsopossible to determine the uncertainty in the estimation {circumflex over(z)}_(k) since the function φ connecting these two estimations is known.The uncertainty in the estimation {circumflex over (z)}_(k) is thenexpressed in the form of the mean square deviationσ_({circumflex over (z)}) _(k) .

At the step 46, other constraints of the set Δ may be defined. Forexample, on the basis of the physical sense of some of the coefficientsof the sets p_(R) and p_(Q), it is possible to set limits on thesecoefficients. For example, if pieces of information on the measurementnoise of the sensor 10 are known, it is possible on the basis on thesepieces of information, to limit the values that can be taken by certaincoefficients of the set p_(R) and p_(Q).

Once the set Δ has been defined, in a step 50, the sensor 32 measuresthe variable z_(k) at N different instants. The number N of instants istypically greater than the number of coefficients to be determinedcontained in the vector p.

Once the variables z_(k) have been measured, they are acquired by thecomputer 34. Then, a step 52, the different variables and coefficientsneeded to minimize the criterion (10) are initialized. For example, theinitial matrix P_(0/0) of estimation of the covariance in the error ofthe estimation {circumflex over (x)}₀ is defined by the followingrelationship:

P _(0/0) =λI,λ>>1

where:

-   -   I is the identity matrix, and    -   λis a constant.

The constant λ is adjusted according to the confidence placed in theinitialization of {circumflex over (x)}₀.

The choice of a constant λ far greater than one means that theuncertainty in the initial estimation {circumflex over (x)}₀ is verygreat.

Here, the initial estimation {circumflex over (x)}₀ and the initialvalue of the coefficients of the vector p are drawn randomly.

Then, a step 54 is performed for determining the vector p whichminimizes the criterion (10). In this step, the values of the vector pare made to vary until a vector p is found that minimizes the criterion(10) and at the same time meets the constraints of the set α. This stepis performed by means of a tool of optimization under constraints. Suchtools of optimization under constraints are known. For example such atool especially is the “Fmincon” function available in the “MATLAB®”development environment, inter alia for digital computation.

Once the vector p which minimizes the criterion (10) has beendetermined, at the step 56 the values of the coefficients of this vectorp are used to calibrate the coefficients of the matrices R and Q of theobserver 18. For example, during this step, the values of thesecoefficients are copied and recorded in the memory 20.

Then, at a step 58, the localizing system 6 uses the coefficientsdetermined by the system 30 to determine the position and orientation ofthe object 4 from the measurements made by the sensor 10.

Many other embodiments are possible. For example, for a same physicalsystem and for a same observer, it is possible to make different choiceson the number of coefficients of this observer that have to bedetermined in order to calibrate it. For example, the number ofcoefficients may be even further reduced by requiring that the differentcoefficients of the diagonal matrix R or Q should all be proportional toa same coefficient. Conversely, the method described here above can alsobe applied to the case where the matrices Q and R are not diagonalmatrices but only positively defined symmetrical matrices. Thisassumption on the contrary increases the number of coefficients to bedetermined by the system 30.

In the criterion (10), the Euclidian norm can be replaced by anothernorm.

Constraints other than those described here above can be used. Forexample, it is possible to require that the variable z_(k) should besystematically or routinely included within the corridor of uncertainty48 at each instant k. This constraint is stronger than the one describedwith reference to FIG. 3 which requires only that the variable z_(k)should on an average be included in the corridor of uncertainty 48.

Different sets of constraints can be used at different instants. Forexample if, at certain instants belonging to a set L, the uncertainty inthe measurement of the variable z_(k) is very low, it is then possibleto use a greater constraint at these instants. For example if theinstant k belongs to the set L, then it is possible to require that thevariable z_(k) should be routinely included in a more limited corridorof uncertainty. For the instants that do not belong to the set L, theconstraint used will be weaker. For example, it will require solelythat, on an average, the variable z_(k) should be included in thecorridor of uncertainty 48. The corridor of uncertainty used to definethe constraint when the instant k belongs to the set L is notnecessarily the same as the one used to define the constraint when theinstant k does not belong to this set L.

Additional constraints other than the ones requiring that the variablez_(k) should be in the corridor of uncertainty of the trajectory of theestimation {circumflex over (z)}_(k) may be used. For example, it ispossible to define constraints that dictate a low variation ofestimation {circumflex over (z)}_(k) between two successive instants.

Choices other than the ones described above are possible forinitializing the optimization of the criterion (10). For example, if theinitial value of the variable x₀ is known, this value is used toinitialize the variable, x₀ and the coefficients of the matrix P₀ areset as a function of the trust in this value.

The uncertainty on the estimation {circumflex over (z)}_(k) may beobtained by means other than the observations of the observer. Forexample, this uncertainty may be determined experimentally or on thebasis of systems of equations other than those defining the observer 18.

The measured variable z_(k) and the measurement y_(k) may coincide. Thusit is not necessary to resort to an additional sensor such as the sensor32.

The calibration method described here above also applies to the ExtendedKalman Filter as well as to the Unscented Kalman Filter (UKF) or anyother recursive filter having parameters to be set. More generally, thecalibration method described here above can be used to calibrate anyobserver of a state variable whose coefficients have to be determinedbeforehand. Thus, the coefficients liable to be determined by means ofthe above method are not limited to those of a covariance matrix.

The invention has been described in the context of a given application.However, it is not limited to this type of application. Otherapplications may be taken into account, such as for example applicationsfor obtaining measurements in the probing of seabeds for which thephysical system makes it possible to have measurements representingsurface/seabed distances or the depths of seas.

1. A method for calibrating coefficients of an observer of a variablestate x_(k) of a physical system from measurements y_(k) of physicalquantities of the system, where the index k identifies the instant ofmeasurement, the method comprising: measuring a variable z_(k) of thephysical system at N different instants, the variable z_(k) being afunction of the state variable x_(k), and determining a vector p ofcoefficients that minimizes the following criterion in complying with apredetermined set Δ of one or more constraints:$\{ {{f(p)} = {\sum\limits_{k = 1}^{N}\; {{z_{k} - ( {{\hat{x}}_{k},p} )}}^{2}}} \} \mspace{20mu} {where}\text{:}${circumflex over (x)}_(k) is the estimation of the variable x_(k) builtby the observer calibrated with the coefficients of the vector p at theinstant k, φ is a known function that links the estimation {circumflexover (x)}_(k) to an estimation {circumflex over (z)}_(k) of the variablez_(k), ∥ . . . ∥² is a norm of the difference between z_(k) and({circumflex over (x)}_(k),p), and the one or more constraints of theset Δ dictate that the trajectory of the variable z_(k) should beincluded in a corridor of uncertainty situated on either side of thetrajectory of the estimation {circumflex over (z)}_(k) at least for themajority of the instants k.
 2. The method of claim 1, wherein the set Δincludes a constraint that dictates that the variable z_(k) should beincluded between {circumflex over (z)}_(k)−a and {circumflex over(z)}k+b, where a and b are predetermined positive constants.
 3. Themethod of claim 2, wherein the constants a and b are proportional to amean square deviation σ_({circumflex over (z)}) _(k) representinguncertainty in the estimation {circumflex over (z)}_(k).
 4. The methodof claim 1, wherein the set Δ of constraints includes a constraint thatdictates that a mean of deviations between the variable z_(k) and itsestimation {circumflex over (z)}_(k) should be below a predeterminedthreshold S₁.
 5. The method of claim 4, wherein the threshold S₁ is afunction of the uncertainty in the estimation {circumflex over (z)}_(k).6. The method of claim 1, wherein the observer is a filter selected froma group consisting of a Kalman filter, an extended Kalman filter, and anUnscented Kalman Filter, and wherein the coefficients to be calibratedare coefficients of the covariance matrices R and Q of the filter. 7.The method of claim 6, wherein the R and Q matrices are chosen to bediagonal matrices.
 8. A computer-readable medium having encoded thereonsoftware for calibrating coefficients of an observer of a variable statex_(k) of a physical system from measurements y_(k) of physicalquantities of the system, where the index k identifies the instant ofmeasurement, the software comprising instructions for: measuring avariable z_(k) the physical system at N different instants, the variablez_(k) being a function of the state variable x_(k), and determining avector p of coefficients that minimizes the following criterion incomplying with a predetermined set Δ of one or more constraints:$\{ {{f(p)} = {\sum\limits_{k = 1}^{N}\; {{z_{k} - ( {{\hat{x}}_{k},p} )}}^{2}}} \}$where: {circumflex over (x)}_(k) is the estimate of the variable x_(k)built by the observer calibrated with the coefficients of the vector pat the instant k, φ is a known function that links the estimate{circumflex over (x)}_(k) to an estimate {circumflex over (z)}_(k) ofthe variable z_(k), ∥ . . . ∥² is a norm of the difference between z_(k)and ({circumflex over (x)}_(k),p), and the one or more constraints ofthe set Δ dictate that the trajectory of the variable z_(k) should beincluded in a corridor of uncertainty situated on either side of thetrajectory of the estimate {circumflex over (z)}_(k) at least for themajority of the instants k.
 9. An apparatus for calibrating coefficientsof an observer of a state variable x_(k) of a physical system frommeasurements y_(k) of physical quantities of the system, where the indexk identifies the instant of measurement, the apparatus comprising: atleast one sensor capable of measuring a variable z_(k) of the physicalsystem at N different instants, the variable z_(k) being a function ofthe state variable x_(k), and a computer capable of determining a vectorp of coefficients that minimizes the following criterion in complyingwith a predetermined set Δ of one or more constraints:$\{ {{f(p)} = {\sum\limits_{k = 1}^{N}\; {{z_{k} - ( {{\hat{x}}_{k},p} )}}^{2}}} \}$where {circumflex over (x)}_(k) is the estimation of the variable x_(k)built by the observer calibrated with the coefficients of the vector pat the instant k, φ is a known function that links the estimation{circumflex over (x)}_(k) to an estimation {circumflex over (z)}_(k) ofthe variable z_(k), ∥ . . . ∥² is a norm of the difference between z_(k)and ({circumflex over (x)}_(k), p), and the one or more constraints ofthe set Δ dictate that the trajectory of the variable z_(k) should beincluded in a corridor of uncertainty situated on either side of thepath of the estimation {circumflex over (z)}_(k) at least for themajority of the instants k.